Write a quadratic equation with the solution set -1 + i√2, -1 - i√2?
To find the quadratic with these roots, we can use the quadratic formula: x = (-b +/- √(b^2 - 4ac)) / 2a, where a, b, and c are coefficients in the standard form of a quadratic equation (ax^2 + bx + c = 0). Substituting -1 +/- i√2 for x, we can solve for a, b, and c to find the quadratic equation with these roots.
Having complex roots means that the quadratic equation does not have any real number solutions. This can happen when the discriminant (b^2 - 4ac) is negative, indicating that the graph of the quadratic does not intersect the x-axis. In this case, the solutions will be imaginary numbers, in the form of a +/- bi, where a and b are real numbers and i is the imaginary unit (√-1).
Yes, the quadratic with these roots can be factored using the quadratic formula. However, the resulting factors will have complex coefficients, which may not be useful in certain situations. It may be more practical to leave the equation in its standard form or use the quadratic formula to find the solutions directly.
To graph a quadratic with complex roots, we can plot the points on the complex plane. The real part of the complex number will correspond to the x-coordinate, and the imaginary part will correspond to the y-coordinate. We can also use the quadratic formula to find additional points on the graph and then connect them to create a smooth curve.
No, there is no shortcut to finding the quadratic with these roots. The most efficient method is to use the quadratic formula, which will give you the exact equation. Other methods, such as completing the square or factoring, may also work but may involve more steps and could result in an approximation rather than the exact equation.